Jump to first page. The Generalised Mapping Regressor (GMR) neural network for inverse discontinuous...

Jump to first page

Jump to first page

The Generalised Mapping Regressor (GMR) neural

network for inverse discontinuous problems

The Generalised Mapping Regressor (GMR) neural

network for inverse discontinuous problems

Student : Chuan LU

Promotor : Prof. Sabine Van Huffel

Daily Supervisor : Dr. Giansalvo Cirrincione

Jump to first page

Mapping Approximation Problem

Feedforward neural networks are : universal approximators of nonlinear continuous

functions (many-to-one, one-to-one) they don’t yield multiple solutions they don’t yield infinite solutions they don’t approximate mapping discontinuities

Jump to first page

Inverse and Discontinuous Problems

Mapping : multi-valued, complex structure.

conditional average of the target data

Poor representation of the mapping by least squares approach (sum-of-squares error function) for feedforward neural networks.

Mapping with discontinuities.

Jump to first page

gatinggatingnetworknetwork

Network 1 Network 2 Network 3

inputinput

outputoutputmixture-of-experts

It partitions the solution between several networks. It uses a separate network to determine the parameters of each kernel, with a further network to determine the coefficients.

winner-take-all

• Jacobs and Jordan• Bishop (ME extension)

kernel blending

Jump to first page

Example #1

ME

MLP

Jump to first page

Example #2

ME

MLP

Jump to first page

Example #3

ME

MLP

Jump to first page

Example #4

ME

MLP

Jump to first page

Generalised Mapping Regressor( GMR )

(G. Cirrincione and M. Cirrincione, 1998)

approximate every kind of function or relation.

input : collection of components of x and y output : estimation of the remaining components output all solutions, mapping branches, equilevel hypersurfaces.

Characteristics :

nm yxyxM :),(

Jump to first page

coarse-to-fine learning incremental competitive based on mapping recovery (curse of dimensionality)

topological neuron linking distance direction

linking tracking branches contours

open architecture

function approximation pattern recognition

Z (augmented) space unsupervised learning

GMR Basic Ideas

clusters mapping branches

Jump to first page

GMR four phases

object merged

Object Merging

Learning Recall-ing

branch 1branch 2

INPUTINPUT

Linking

links

object 1

pool of neurons

object 2object 3

TrainingTrainingSetSet

Jump to first page

EXIN Segmentation Neural Network (EXIN SNN)

clustering

(G. Cirrincione, 1998)

w4= x4

vigilance threshold

x

Input/weight space

Z (augmented) space

coarse quantization• EXIN SNN• high z ( say 1 )

branch (object)neuron

GMR Learning

Z (augmented) space

• production phase• Voronoi sets domain setting

GMR Learning

Z (augmented) space

• secondary EXIN SNNs• z = 2 < 1

TS#1

TS#2

TS#3

TS#4

TS#5

Other levels are possible

fine quantization

GMR Learning

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1PLN Level 1 1=0.2, epoch1=3

x

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1PLN Level 1 1=0.2, epoch1=3

x

GMR Coarse to fine Learning ( Example)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1PLN level 1-2, 1=0.2, epoch1=3;2=0.1, epoch2=3

* 1st PLN: 13*

x

y

* 2nd PLN: 24*

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1PLN level 1-2, 1=0.2, epoch1=3;2=0.1, epoch2=3

* 1st PLN: 13*

x

y

* 2nd PLN: 24*

object neuron

fine VQ neurons

object neuron Voronoi set

Jump to first page

GMR Linking Voronoi set: setup of the neuron radius (domain variable)

neuron i

ri

asymmetric radius

Task 1 :Task 1 : Task 1 :Task 1 :

Jump to first page

Weight Space

GMR Linking For one TS presentation:

zi

d1

w1

w5

w3

w4

d1

w2

d5

d3

d4

d2

branch and bound search technique

k-nn

Linking candidates

distance test direction test create a link or strengthen a link

Task 2 :Task 2 : Task 2 :Task 2 :

Linking direction

Jump to first page

Branch and Bound Accelerated Linking

neuron tree constructed during learning phase (multilevel EXIN SNN learning)

methods in linking candidate step (k-nearest-neighbors computation): -BnB : < d1 , ( : linking factor predefined) k-BnB : k predefined.

Jump to first page

44 43

3127

6459

5547

7681 80 83

0,00%10,00%20,00%30,00%40,00%50,00%60,00%70,00%80,00%90,00%

2-D (TS 2k): 8 2-D(TS 4k): 24 3-D (TS 3k): 199 linking flops (x100,000)

percents of linking flops saved by branch and bound

2-level d-BnB

2-level k-BnB

3-level d-BnB

3-level k-BnB

GMR Linking

branch-and-bound in linking experimental results:83 %

Jump to first page

branch and bound (cont.)

Apply branch and bound in learning phase ( labelling ) :

Tree construction k-means EXIN SNN

Experimental results (in the 3-D example) 50% of labeling flops are saved

GMR Linking Example

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Linking: = 2.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Linking: = 2.5

link

GMR Merging Example

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Merging: threshold = 1

Obj: 13 -> 3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Merging: threshold = 1

Obj: 13 -> 3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

x = 0.2

Level 1 neurons: 3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

x = 0.2

Level 1 neurons: 3

GMR Recalling Example

)04.0

01.0)2(sin(

4

1)(

2

xxxfy )

04.0

01.0)2(sin(

4

1)(

2

xxxfy

level 1 neuron

level 2 neuron

branch 1

branch 2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

y = 0.6

Level 1 neurons: 1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

y = 0.6

Level 1 neurons: 1

level one neurons : input within their domain level two neurons : only connected ones level zero neurons : isolated (noise)

Experiments

spiral of Archimedes = a (a = 1)

spiral of Archimedes = a (a = 1)

Experiments

Sparse regions

further normalizing + higher mapping resolution

)04.0

01.0)2(sin(

4

1)(

2

xxxfy )

04.0

01.0)2(sin(

4

1)(

2

xxxfy

Experiments noisy data

1 Bernoulli of lemniscate

222222

a

yxayx 1 Bernoulli of lemniscate

222222

a

yxayx

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Solutions for y = -0.5

Level 1 neurons: 6

Experiments

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Solutions for y = -0.1

Level 1 neurons: 10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Solutions for y = 0.5

Level 1 neurons: 5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

y

Solutions for y = 1

Level 1 neurons: 19

5,3 Lissajous of curve

sin,cos

ba

btyatx 5,3 Lissajous of curve

sin,cos

ba

btyatx

Experiments

contours : links among

level one neurons

GMR mapping of 8 spheres in a 3-D scene.

Jump to first page

Conclusions

GMR is able to : solve inverse discontinuous problems approximate every kind of mapping

yield all the solutions and the corresponding branches

GMR can be accelerated by applying tree search techniques

GMR needs : interpolation techniques kernels or projection techniques for high dimensional data adaptive parameters

Jump to first page

Thank you !(shi-a shi-a)

l1 = 0b1 = 0

l1 = 0b1 = 0

l6 = 0b6 = 0

l6 = 0b6 = 0

l5 = 0b5 = 0

l5 = 0b5 = 0

l2= 0b2= 0

l2= 0b2= 0

l3 = 0b3 = 0

l3 = 0b3 = 0 l4 = 0

b4 = 0

l4 = 0b4 = 0

l7 = 0b7 = 0

l7 = 0b7 = 0

l8= 0b8 = 0

l8= 0b8 = 0

l3 = 2b3 = 1

l3 = 2b3 = 1

GMR Recall

input

w1

w2

w3

w7

w8

w4

w5

w6

r1

l1 = 1b1 = 1

l1 = 1b1 = 1

linking tracking

restricted distance

level one test

connected neuron :level zero level two

branch the winner branch

GMR Recall

input

w1

w2

w3

w7

w8

l1 = 0b1 = 0

l1 = 0b1 = 0

l6 = 0b6 = 0

l6 = 0b6 = 0

l5 = 0b5 = 0

l5 = 0b5 = 0

l2= 0b2= 0

l2= 0b2= 0

l3 = 0b3 = 0

l3 = 0b3 = 0 l4 = 0

b4 = 0

l4 = 0b4 = 0

l7 = 0b7 = 0

l7 = 0b7 = 0

l8= 0b8 = 0

l8= 0b8 = 0

w4

w5

w6

r2

l1 = 1b1 = 1

l1 = 1b1 = 1

l3 = 2b3 = 1

l3 = 2b3 = 1

l2= 1b2= 2

l2= 1b2= 2 l2= 1b2= 1

l2= 1b2= 1

level one test

linking tracking

branchcross

GMR Recall

l6 = 0b6 = 0

l6 = 0b6 = 0 l6 = 2b6 = 4

l6 = 2b6 = 4 l6 = 1b6 = 6

l6 = 1b6 = 6

input

w1

w2

w3

l1 = 0b1 = 0

l1 = 0b1 = 0

l5 = 0b5 = 0

l5 = 0b5 = 0

l2= 0b2= 0

l2= 0b2= 0

l3 = 0b3 = 0

l3 = 0b3 = 0 l4 = 0

b4 = 0

l4 = 0b4 = 0

l7 = 0b7 = 0

l7 = 0b7 = 0

l8= 0b8 = 0

l8= 0b8 = 0

w4

w5

w6

l1 = 1b1 = 1

l1 = 1b1 = 1

l3 = 2b3 = 1

l3 = 2b3 = 1

l2= 1b2= 2

l2= 1b2= 2 l2= 1b2= 1

l2= 1b2= 1

l4 = 1b4 = 4

l4 = 1b4 = 4

l5 = 2b5 = 4

l5 = 2b5 = 4 l4 = 1b4 = 5

l4 = 1b4 = 5 l4 = 1b4 = 4

l4 = 1b4 = 4

… until completion of the candidates

level one neurons : input within their domain level two neurons : only connected ones level zero neurons : isolated (noise)

w7

w8

l6 = 1b6 = 4

l6 = 1b6 = 4

clipping

Tow Branches

Tow Branches

Two Branches

Two Branches

GMR Recall

input

w1

w2

w3

w7

w8

l7 = 0b7 = 0

l7 = 0b7 = 0

l8= 0b8 = 0

l8= 0b8 = 0

w4

w5

w6

Output = weight complements of the level one neurons Output interpolation

l1 = 1b1 = 1

l1 = 1b1 = 1

l3 = 2b3 = 1

l3 = 2b3 = 1

l2= 1b2= 1

l2= 1b2= 1

l4 = 1b4 = 4

l4 = 1b4 = 4

l4 = 1b4 = 4

l4 = 1b4 = 4 l6 = 1

b6 = 4

l6 = 1b6 = 4